Course 2E01 (Frolov), Multivariable Calculus
Tutorial Sheet 6
Due: at the end of the tutorial session Tuesday/Thursday, 15/17 November 2011
1.
(a) Express rectangular coordinates in terms of spherical coordinates.
Draw the corresponding picture.
(b) Consi
Course 2E01 (Frolov), Multivariable Calculus
Tutorial Sheet 5
Due: at the end of the tutorial session Tuesday/Thursday, 1/3 November 2011
1. Consider the portion of the surface (y 1)2 + (z 1)2 = 12 that is above the rectangle
R = cfw_(x, y) : 3 x 3 , 2 y
Course 2E01 (Frolov), Multivariable Calculus
Tutorial Sheet 8
Due: at the end of the tutorial session Tuesday/Thursday, 29 November/1 December 2011
1. Consider the lamina that is the portion of the surface x2 + y 2 6z + 1 = 0 inside the
surface x2 + y 2 =
Course 2E01 (Frolov), Multivariable Calculus
Tutorial Sheet 7
Due: at the end of the tutorial session Tuesday/Thursday, 22/24 November 2011
1. Consider the surface that extends upward from the semicircle y =
plane to the surface
y2
1
z = x2 y +
12
3
(a) M
Course 2E01 (Frolov), Multivariable Calculus
Tutorial Sheet 10
Due: at the end of the tutorial session Tuesday/Thursday, 14/16 December 2011
1.
(a) Solve the following initial value problem by the Laplace transform
y + 9 y = 9 u(t ) + (t 3) ,
y(0) = 1 ,
y
January 25, 2010
Lecturer Dmitri Zaitsev
Hilary Term 2010
Course 2E02 2010 (SF Engineers & MSISS & MEMS)
Sheet 1
Due: at the end of the tutorial
Exercise 1
Find v + u, 2v, kuk, kvk, the dot product u v, the angle between u and v and
determine whether u an
February 15, 2011
Lecturer Dmitri Zaitsev
Hilary Term 2011
Course 2E02 2011 (SF Engineers & MSISS & MEMS)
Sheet 4
Due: at the end of the tutorial
Exercise 1
Find the coordinates of the vector v with respect to the basis v1 , . . . , vn (i.e. the coefficie
February 22, 2010
Lecturer Dmitri Zaitsev
Hilary Term 2011
Course 2E02 2011 (SF Engineers & MSISS & MEMS)
Sheet 5
Due: at the end of the tutorial
Exercise 1
Find the rank and the nullity of the matrix:
(i) ( 2 1 1 );
2 1 1
(ii) 1 1
1 .
1 2 2
Exercise 2
Ca
February 1, 2011
Lecturer Dmitri Zaitsev
Hilary Term 2011
Course 2E02 2011 (SF Engineers & MSISS & MEMS)
Sheet 2
Due: at the end of the tutorial
Exercise 1
Find T (x) = Ax for the matrix A and the vector x whenever the product makes sense
(i.e. the sizes
March 22, 2010
Lecturer Dmitri Zaitsev
Hilary Term 2011
Course 2E02 2011 (SF Engineers & MSISS & MEMS)
Sheet 7
Due: at the end of the tutorial
Exercise 1
Find the characteristic polynomials of the following matrices:
0 5
(ii)
;
1 0
1 1 1
(iii) 0 1 2 ;
0 0
March 29, 2010
Lecturer Dmitri Zaitsev
Hilary Term 2011
Course 2E02 2011 (SF Engineers & MSISS & MEMS)
Sheet 9
Due: at the end of the tutorial
Exercise 1
Find the Fourier series of the function
f (x) =
1 if x < 0
,
2 if 0 x ;
x .
Exercise 2
Identify even
March 8, 2011
Lecturer Dmitri Zaitsev
Hilary Term 2011
Course 2E02 2011 (SF Engineers & MSISS & MEMS)
Sheet 6
Due: at the end of the tutorial
Exercise 1
Calculate the coordinates of v relative to the orthogonal basis
cfw_(1, 0, 0) , (0, 2, 1) , (0, 1, 2)
February 7, 2010
Lecturer Dmitri Zaitsev
Hilary Term 2011
Course 2E02 2011 (SF Engineers & MSISS & MEMS)
Sheet 3
Due: at the end of the tutorial
Exercise 1
(i) Find parametric equations for the line spanned by the vector:
u = (1, 2, 5);
(ii) Give two equa
UNIVERSITY OF DUBLIN XMA2E021
TRINITY COLLEGE
FACULTY OF ENGINEERING, MATHEMATICS
AND SCIENCE
SCHOOL OF MATHEMATICS
JF Engineers Trinity Term 2011
JF MSISS
JF MEMS
COURSE: MA2EO2 — ENGINEERING MATHEMATICS IV
Saturday, May 7 GOLDHALL 9:30 —« 11:30
Dr. D. Z
Course 2E1 (Frolov), Multivariable Calculus
Tutorial Sheet 11
Due: at the end of the tutorial session Wednesday/Thursday, 4/5 February 2009
Name:
1. Find the mass of the lamina with the constant density 0 that is the portion of the
paraboloid 2z = x2 + y
Course 2E1 (Frolov), Multivariable Calculus
Tutorial Sheet 2: Solutions
Due: at the end of the tutorial session Tuesday/Thursday, 11/13 October 2011
1. Sketch the level curve z = k for the specied values of k
z = x2 + 2x + 4y 2 + 4y ,
k = 2, 1, 2 .
Soluti
UNIVERSITY OF DUBLIN
XMA2E011
TRINITY COLLEGE
Faculty of Engineering, Mathematics
and Science
school of mathematics
SF Engineers
SF MSISS
SF MEMS
Trinity Term 2011
Module 2E011,
?, ?
Final Exam
?
9.30 11.30
Dr. Sergey Frolov
ATTEMPT QUESTION 1 and FOUR OT
Course 2E1 (Frolov), Multivariable Calculus
Tutorial Sheet 14
Due: at the end of the tutorial session Wednesday/Thursday, 25/26 February 2009
Name:
Solve the following initial value problems by the Laplace transform. Sketch or plot the input
function and
Course 2E1 (Frolov), Multivariable Calculus
Tutorial Sheet 15
Due: at the end of the tutorial session Wednesday/Thursday, 4/5 March 2009
Name:
1. Applying convolution, nd and sketch or plot the solution. (Show the details of your
work.)
1 if 0 < t < 1
y +
Course 2E1 (Frolov), Multivariable Calculus
Tutorial Sheet 12
Due: at the end of the tutorial session Wednesday/Thursday, 11/12 February 2009
Name:
1. Consider the vector eld
F(x, y, z) = (3x + 2z 2 ) i (2y + 3x3 ) j + (5z + y 3 ) k .
(a) Use the Divergen
Course 2E1 (Frolov), Multivariable Calculus
Tutorial Sheet 13
Due: at the end of the tutorial session Wednesday/Thursday, 18/19 February 2009
Name:
Solve the following initial value problems by the Laplace transform. (Show the details of
your work.)
1.
y
UNIVERSITY OF DUBLIN
XMA2E011
TRINITY COLLEGE
Faculty of Engineering, Mathematics
and Science
school of mathematics
SF Engineers
SF MSISS
SF MEMS
Trinity Term 2010
Module 2E011
Tuesday, April 27
LUCE LOWER
9.30 11.30
Dr. Sergey Frolov
ATTEMPT FIVE QUESTIO
Course 2E1 (Frolov), Multivariable Calculus
Tutorial Sheet 3
Due: at the end of the tutorial session Tuesday/Thursday, 18/20 October 2011
1. Consider the function
y 2 + z x + 2 cos(3x 2z) ,
f (x, y, z) =
and the point P (2, 1, 3) .
(a) Find a unit vector